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Apportionment Show #2 of 7 Message to the user... The most effective way to use a PowerPoint slide show is to go to “SLIDE SHOW” on the top of the toolbar, and choose “VIEW SHOW” from the pull down menu. OR, using the shortcut toolbar on the bottom left, choose the rightmost icon (“SLIDE SHOW”) Use the spacebar, enter key or mouse to move through the slide show. Use the backspace key to undo the last animation on a slide TEACHERS: If using this show as part of a lecture, it is helpful to go to “PRINT” in the “FILE” menu and use the drop down menu at the bottom left: “PRINT WHAT.” Printing the “OUTLINE VIEW” will be helpful if you intend to view many slides with your class; or you can print a particular slide to use as a handout. (Many shows will include sound… you may want to turn on your speakers!)

Hamilton’s Method Apportionment Vocabulary & Example Hamilton’s Method of Apportionment --A bundle of contradictions??

Because apportionment applications so often deal with representative governments, the vocabulary for the “generic” apportionment application will use words that relate to that type of problem. When we do specific applications, we will make sure to use the applicable terms. For example, in generic terms, we will be apportioning “seats” to particular “states” based on their “populations” Whereas in a specific application the “seats” may be canned goods the “states” may be charity organizations and the “populations” may be the number of people served per day. Hamilton’s Method: Vocabulary

We will begin these applications by creating a chart with 6 columns Label the columns as seen here... The number of rows will depend on the number of “states”. State population SQ LQ Rank Apmt Hamilton’s Method: Vocabulary & Example The ORANGE type indicates that these columns will be labeled appropriately for the given application. apportionment

APPLICATION: Mathland is a small country that consists of 3 states; Algebra, Geometry, and Trigonometry. The populations of the 3 states will be given in the chart. Suppose that one year, their government’s representative body allows 176 seats to be filled. The number of seats awarded to each state will be determined using Hamilton’s Method of apportionment. Hamilton’s Method: Vocabulary & Example

*handout State population SQ LQ Rank #reps Algebra 9230 Geometry 8231 Trig 139 TOTAL: #seats 176 SD: Hamilton’s Method: Vocabulary & Example

*handout State population SQ LQ Rank #reps Algebra 9230 Geometry 8231 Trig 139 TOTAL: #seats 176 SD: Hamilton’s Method: Vocabulary & Example Print out previous slide or create this chart before going on to next slide...

Use the given information to calculate the TOTAL population and the “standard divisor” The Standard Divisor is the “number of people per representative” [calculated by dividing TOTAL by #seats] Here, the SD means that for every 100 people a state will receive 1 representative State population SQ LQ Rank #reps Algebra 9230 Geometry 8231 Trig 139 TOTAL: 17600 #seats 176 SD: 100 Hamilton’s Method: Vocabulary & Example

The STANDARD QUOTA is the exact quotient upon dividing a state’s population by the SD The SQ is the exact # of seats that a state would be allowed if fractional seats could be awarded. Notice that the “# of seats awarded” can also be thought of as the “# of representatives” for a state. State population SQ LQ Rank #reps Algebra 9230 Geometry 8231 Trig 139 TOTAL: 17600 #seats 176 SD: 100 Hamilton’s Method: Vocabulary & Example

The STANDARD QUOTA is the exact quotient upon dividing a state’s population by the SD For example… The SQ for Algebra is: 9230/100 = 92.3 The SQ for Geometry is: 8231/100 = 82.31 etc… You will note that, very often, the values upon division must be rounded. There is a danger in rounding to too few decimal places, as the decimal values will be used to determine which states are awarded the remaining seats. State population SQ LQ Rank #reps Algebra 9230 Geometry 8231 Trig 139 TOTAL: 17600 #seats 176 SD: 100 Hamilton’s Method: Vocabulary & Example 92.3 82.31 1.39

The LOWER QUOTA is the integer part of the SQ For example… The LQ for Algebra is: 92 etc… It may be thought of as “rounding down” to the lower of the two integers the SQ lies between. State population SQ LQ Rank #reps Algebra 9230 92.3 Geometry 8231 82.31 Trig 139 1.39 TOTAL: 17600 #seats 176 SD: 100 Hamilton’s Method: Vocabulary & Example 92 82 1

If each state was awarded its LQ, What would the total # of seats apportioned be? That leaves 1 seat empty! And we certainly can’t have that! State population SQ LQ Rank #reps Algebra 9230 92.3 92 Geometry 8231 82.31 82 Trig 139 1.39 1 TOTAL: 17600 #seats 176 SD: 100 Hamilton’s Method: Vocabulary & Example 175

The 1 empty seat will be awarded by ranking the decimal portions of the SQ. Since we only have 1 seat to fill, we need only rank the highest decimal portion. State population SQ LQ Rank #reps Algebra 9230 92.3 92 Geometry 8231 82.31 82 Trig 139 1.39 1 TOTAL: 17600 #seats 176 SD: 100 Hamilton’s Method: Vocabulary & Example 1st 175

So, in this application, the state with the highest decimal portion is awarded its UPPER QUOTA The Upper Quota is the first integer that is higher that the SQ. It may be thought of as “rounding up” to the higher of the two integers the SQ lies between. And now the TOTAL # reps is equal to the allowable #seats! State population SQ LQ Rank #reps Algebra 9230 92.3 92 Geometry 8231 82.31 82 Trig 139 1.39 1 TOTAL: 17600 #seats 176 SD: 100 Hamilton’s Method: Vocabulary & Example 92 82 1st 2 175 176

In practice, Hamilton’s Method of apportionment is very straightforward. In fact, for one-time applications, it’s probably the easiest method to use. The only confusion might occur if two decimal values are the same and both can not be ranked. In that case, usually the state with the higher integer value would be awarded the extra seat. Hamilton’s Method seems simple enough... So, what’s the problem?

Problems may occur when Hamilton’s Method is used repeatedly in an application which has certain changes (in # of seats, populations, and/or # of states) over a period of time. In practice, these problems rarely occur. But, when they do, they create controversies about the “fairness” of the re-apportionment. In fact, an apportionment method equivalent to Hamilton’s was actually used for awhile in the USHR… But the discovery of these problems caused it to be abandoned. Hamilton’s Method seems simple enough... So, what’s the problem?

Going on?... Apportionment: Show #3: Hamilton’s Method -- Alabama Paradox End of show #2 Prepared by Kimberly Conti, SUNY College @ Fredonia Suggestions and comments to: Kimberly.Conti@fredonia.edu

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